Math tools and strategies for today's standards…that lay the foundation for tomorrow!

Author: aerecord

This year, our school has chosen to work on addition and subtraction word problem types in our grades 3-5. You may think that it seems strange to work on add/sub in the upper grades, but I assure you it is quite a complicated endeavor. Read my previous post on the various problem types here: Word problems with addition and subtraction are easy, right?

Our initiative exposes all of our students to the variety of addition/subtraction word problem types (then we’ll go on to mult/div) as based on the research in the book Cognitively Guided Instruction (and described in the CCSS) and have used the tape diagram as the way to model these situations. We have followed the rectangular tape diagram as found on Greg Tang’s math website within his word problem generator program. (If you haven’t checked out this part of his site, I would encourage you to do so. You can select the problem type, the unknown, and the number range. Awesome resource to create differentiated Guided Math workstations). There is a large rectangle on top and beneath it are two smaller boxes whose size equals the large rectangle and hence the values of the two smaller boxes must equal the value in the top box. I find it an incredibly powerful model that allows us to teach many mathematical concepts. Here is an example from the site at http://www.gregtangmath.com.

First, the tape diagram provides us a visual model to teach the relationship between addition and subtraction since the values of both boxes (and eventually more when there are more addends introduced) must equal the value of the large box. I love using Cuisenaire rods to model this using numbers within 10 – even for our upper elementary students since our goal initially is not about the computational skills to get the numerical answer, but the underlying structure of the situation. If the top box is unknown, students will need to add the two smaller boxes. If one of the smaller boxes is unknown students have a choice of adding OR subtracting. There is no dictate in terms of how they can find the numerical answer. The problem may be an “Add to” problem type, but they could use subtraction to solve it if one of the addends is unknown.

For example, there were 5 more sheep than cows on a farm. There were 7 sheep. How many cows were there?

Before my work with CGI and the tape diagramming it would take me a bit to figure out exactly what is going on. Now, though, I know that there is a bigger amount and a smaller amount being compared as well as a difference between those two. I actually see the tape diagram in my head. After I read the first sentence, I know one of the smaller boxes is the difference of 5 and I can even label the small box cows the larger box sheep since I know there are more sheep than cows. Once I read that there are 7 sheep, I would put that value in the big box and then be left to my own decision as to how I want to compute the answer of how many cows there are. For example, I may want to use 7-5 or I may want to think 5 + ? = 7. That’s where the student choice comes in. When numbers become larger, students would then choose if they want to use an open numberline, act out with blocks, use a 100’s chart, draw a picture, etc. Total freedom to figure out the numerical answer, yet the model is the same.

Thirdly, these problem types will follow our students throughout many years of their math journey. It’s just that the numbers will get larger or become fractions and decimals. Yet the underlying structure is always there. If we teach the tape diagram from the earliest years, I feel like we are providing them with a tool in understanding the structure for years to come. They will then bring to the table their own inventive strategies for calculating the answers.

Finally, having learned this particular model for the add/sub problem types, students will have developed the mathematical thinking to find answers where they may not need to find a numerical answer at all. I’ve seen many questions on our district standardized testing measure NWEA as well as SBAC where it gives the students the situation and then asks them to choose an equation that could be used to solve it. There’s no number answer at all. Inherent in the given equation choices are understandings of the commutative property, the equal sign as meaning “the same as”, the inverse relationships between addition and subtraction, etc. All powerful foundational mathematics concepts we are able to discuss while working on this…and the exact same model is used for all the add/sub problem types!

So, while I’m not in favor usually of mandating things, we are mandating the model including labeling the diagrams with words from the word problem, but in no way are we mandating the computational methods. In fact, in many instances we don’t find the answers at all. We have students determine which equations, of many given, could be used to solve the problem. We put the answers on the left side of the equal sign, we change the order of the addends, we have some with the inverse operation, and we even have some with the smaller number minus the larger number to discuss that we COULD do that math but we’d end up in negative numbers and not find the answer we are looking for (we live in a cold area so weather below zero is very familiar to them). Students are doing a lot of discussing about why given equations will solve the situation as well as why some will not. I get the chills listening to them support their answers.

I have struggled with the idea of “mandating” as well, but I feel like the tape diagram specifically provides such rich mathematical concepts that it is worth it. I’m not seeing nearly as much as I used to of students mindlessly plucking numbers from word problems and finding a key word that would suggest an operation with no thought to the context in the problem. I think that right there is a win.

I’m hoping that in this process, not only are we setting the foundation of the structure of addition/subtraction word problems but we are encouraging the thinking process needed to diagram word problem situations that we can then apply to more difficult problem types they will encounter down the road like multiplication and division of whole numbers, decimals and fractions.

Over the course of my 13 years as a grade 5 teacher and now in my third year as a Math Coach, I have witnessed the same situation over and over….a student is presented with a word problem, their brain freezes and they say, “I don’t get it.” The more I work with the problem types in the CCSS (which are based on Thomas P Carpenter, Elizabeth Fennema, Megan Loef Franke and Linda Levi’s Cognitively Guided Instruction) as well as the modeling of these word problems as found in the word problem generator on Greg Tang’s website, the more I am convinced they are key to helping us with this age-old problem and will even set the foundation for future math learning! So, so powerful!

For years I thought I was helping my students successfully answer word problems by having them highlight the numbers and the key words that help them decide what operation to do…. I even had the most beautiful key word poster hanging in my classroom! My students were mostly successful because the vast majority of the word problems they were asked to solve had the result as the unknown. Now, though, we introduce word problems starting in grade 1 where the unknown could be at any of the places and key words simply will not work anymore.

Here’s an example:

There were 6 birds in a tree. Some more flew in. Now there are 9 birds in the tree altogether. How many birds flew in?

More students than we like would see the numbers 6 and 9 and the word “altogether” and think they need to add them together. They would give the answer of 15 and not look back.

At our school this year, we are dedicating one day of our math intervention time to working on these word problem types and the objective is NOT to find the right numerical answer. In fact, in my grade 3-5 classrooms I have asked that we only use numbers within 10 to start because I don’t want the cognitive demands of the computation to get in the way of the underlying structure of the word problems. What I am expecting students to do is:

make a model of the word problem with Cuisenaire rods (providing the concrete representation…until they don’t need to see it anymore)

draw the model and label it with words from the word problem, the numbers we know, and to put a question mark for the unknown

write at least 2 equations that could be used to solve the problem (keeping in mind two important mathematical concepts: the commutative property of addition as well as the understanding that an equal sign means the “same as” and is a relational symbol, not an action symbol)

write at least 2 equations that could NOT be used to solve the problem (Opens up the discussion that there is no commutative property of subtraction…if you write 4-9 it will end up in negative numbers. We live in northern NH so our temperature frequently goes below zero which helps my students understand this.)

Here’s are some posters I made to help our staff as we work with our students on this model.

As you can see, the model will work for any addition or subtraction word problem type the students will come across. It also perfectly matches the Cuisenaire concrete model when the numbers are kept within 10. As students progress in their math journey, the numbers will be within 100…within 1,000…and eventually include decimals or fractions…but the basic structure is exactly the same!

If you haven’t checked out Greg Tang’s word problem generator you need to go there right now and check it out at http://www.gregtangmath.com. Absolutely phenomenal! You can generate word problems of a specific problem type, specify what you want to be unknown, and determine the number range. You have the choice of generating one problem or ten that you can then print out. In addition, when you click on the “hint” it will show you the model which will help you decide how you want to solve the problem.

We have given a pretest to all our kiddos with one question for each problem type with the unknowns at each spot so we can zoom in on exactly the word problem type each student needs to begin working on. I’m hopeful that it won’t be too long before students see a word problem and say, “I can figure this out!”

I am in my second year of being a Math Coach K-5 working in 23 classrooms, but before this position I was a grade 5 teacher in a self-contained classroom for 13 years. While I loved teaching all subjects, Math was always my favorite part of the day. As the years went on, I found it increasingly difficult and frustrating teaching Math because the ability spectrum among the students in my classroom was so huge and just seemed to get wider and wider. I felt that my whole-group lessons were really only meeting the needs of 1/3 of my class. After a lesson, 1/3 were still lost and 1/3 had already known what I taught and were bored.

I am eternally grateful for two things I learned about that literally transformed my classroom: Guided Math and the math app Front Row.

Guided Math

I had the life-changing opportunity to attend a session at a workshop led by Dr. Nicki Newton. She is an absolute wealth of knowledge of teaching math. (go on to Youtube and search for Dr. Nicki – you will find so many fabulous videos explaining guided math as well as focus priorities for elementary grades as well as fluency activities for elementary gradelevels). Her first passion was literacy but then found herself absorbed in math education. She brilliantly applied her knowledge of literacy education to math! Guided Reading had been done for years in literacy but she was the first person I came across (I’m sure there are others out there -I just wasn’t aware of them) that applied it to Math instruction. (She is also coming out with a book this summer on Math Running Records which has transformed my life – again!) Dr. Nicki introduced me to the idea of math workstations and differentiating my instruction within a Guided Math time with my students. It truly transformed my classroom!! Here are her two books she has written on the topic:

At the beginning of my journey with this format for math instruction, students rotated through three workstations during my hour of math. They spent part of the time with me learning the topic of the day tiered to their ability level, part of the time doing independent work, and then part of the time on a phenomenal math app called Front Row.I felt like I was able to meet the needs of each of my students (from the struggling ones to the advanced ones) in a way I was never able to before!

Now that I am in a position of Math Coach, I am able to focus on helping other teachers implement this format for math time. Classrooms typically have 3 or 4 stations depending on the age of the students. One station is usually math fact fluency (NOT focusing on speed but on strategies which is an entire blog post on its own!) through playing games, a technology station (mostly Front Row (see below) but also Greg Tang apps or other IXL), an independent station practicing a skill that has already been taught, and a teacher station geared to what the students need. It is so exciting to watch in action!!

Front Row – the nirvana of math apps!

I had already been implementing this structure of teaching for a little while when I saw a post on Facebook by Scholastic that there was a new free K-8 math app called Front Row. When I looked into it I was so thrilled to see that within the app they had created features that had been taking me hours of time to do by hand!

Here are just a few of the powerful features:

setting up the classroom was so easy – I just went to http://www.frontrowed.com and created a class…then I just needed to type in my students’ names…that’s it!

completely individualized for each student – students just take a benchmark test for each domain starting at kindergarten level questions and the program decides which skill they need to begin working on

when the students are working on a question, there are a bunch of virtual manipulatives at their fingertips to use to help them figure out the answer

there is button on the side for students to watch a video that teaches them the skill

if students get a question wrong, the program will suggest other students in the classroom that can help them since it knows which level all students are working on (this was one of the most magical moments for me – watching students help other students)

the teacher features were amazing –

at a click of a button I would get groupings of same-ability or mixed-ability…for the same-ability the program showed exactly which skill they all needed to work on

report cards for each student

homework sheets individualized for the skills each student needed – yet I only had to press “print”

And that was two years ago…Front Row has only become more powerful! Here are some added features now:

addition of a whole Reading component

library of Inquiry Based Lessons

Math fact fluency (that is NOT based on speed!!!! yea!)

question styling that includes multiple answers, dragging many answers to an appropriate column such as less than one, equal to one or more than one….truly getting to the heart of a students’ conceptual understanding rather than just multiple choice.

One of the best parts is that all of the above features are completely free! There is a school edition that costs money (a couple of schools in our district have chosen to pay for this since we are finding it to be so powerful) but all of the features above are in the free version.

Now that I’m teaching preservice teachers, I have also included this as part of my assignements for our class. That way, I can be assured that my college students have any gaps in their knowledge of foundational skills filled in. They have the entire semester to complete domains that cover the content of my course. That way it is at their own pace and there are teaching videos right there for them. As the teacher, I can keep tabs on them and help them when needed as well. So powerful!!!

Disclaimer: I do not benefit financially from Front Row. One of the founders of the app, Sidharth Kakkar, was so kind and genuinely interested in getting feedback from teachers two years ago that I felt very comfortable emailing him with any questions or concerns we were having. We developed a wonderful rapport. Over time, he asked me if I would mind being interviewed by journalists covering this growing company about my thoughts on Front Row…including an interview for an article by CNN! As the company has grown, they have developed an Ambassador program made up of teachers who love Front Row. I’m so honored to be one of them. As a special treat and privilege, I will be presenting on their behalf at a Share Fair Nation conference in South Carolina next month. So incredibly thankful for the opportunity to share my enthusiasm for this amazing app!

When I reflect on one thing that I think has the potential to change the math journey for every child, I think of Number Talks! I first learned about Number Talks when I purchased Sherry Parrish’s Number Talks book. I loved that it included example dot patterns, 10frame patterns, rekenrek patterns, and examples for every operation that I could use in my classroom as well as a DVD showing students actually doing number talks from kindergarten to 5th grade. I was floored with their thinking! There were some times I had to pause the video to try to figure out what the children were doing to solve some pretty sophisticated expressions! Wow!

I also love the book Making Number Talks Matter by Cathy Humphreys and Ruth Parker because it not only discusses the strategies that students employ but it also shows how the very same strategies used for simple expressions are applied down the road to larger numbers, decimals, and fractions! So incredibly powerful!!

Here are the reasons why I love Number Talks so much:

students are actively constructing their own thinking in a way that makes sense to them

number talk routines get children talking about numbers and their thinking – so many times I have learned about a way of solving a problem that I never would have thought of on my own!

students learn that there are many ways to solve a problem (not just one as those who are taught with the algorithm come to believe) and respect the perspectives of others

students learn from each other and even learn to critique the work of others as they explain their strategies

working on thinking strategically through number talks unlocks the ability to solve problems children would never have previously thought they could! I have asked children who have memorized their math facts to solve 4 x 12 and they have told me, “I haven’t learned that one yet.” When I ask students who have had number talks to solve 4 x 12, they immediately begin thinking about how they can figure it out…and they have a couple of ways of doing it!

students are given the opportunity to make connections between topics that would never have happened when focusing only on an algorithmic way of solving problems….one 4th grade classroom I work in was working on the angles within a circle. When we did a number talk for 45 x 16, one student answered 720 pretty quickly. He explained that since he knew there were 8-45 degree angles in a circle, 16 would be 2 circles worth of degrees. Amazing!

number talks allow me to talk to students about having a growth mindset (I love Carol Dweck and Jo Boaler’s work on this!)….students know that if they make a mistake when trying out a new strategy they are growing their brains!

students can enter the activity at their readiness level – if I give 7 + 8, students who are still counting on can solve it…but it also allows students who realize a doubles plus/minus 1 strategy could be used or even students who take some away from the other number to make a 10 are all successful and respected!

When most people are asked how to define fluency, they think of speed and accuracy. I think everyone would agree that students who have automatized their math facts are in a much better position to learn more difficult concepts because they have more brain energy to concentrate on understanding the concept they are learning rather than worrying about the math facts embedded in the problem. There is a third part to this definition, though, that is the key to laying the foundation for future math learning – flexibility! When children are taught strategies that allow them think flexibly about numbers, we are helping them develop their number sense and unlocking students’ access to more difficult concepts later. So not only are we helping them learn the math facts, but those very same strategies can be applied to larger numbers, and even fractions and decimals down the road.

In our district, we have been following a progression of strategies for each operation developed by Dr. Nicki Newton in her upcoming book Running Records due out in the spring (2016). I took a webinar this past summer to learn about this instructional assessment tool and it has simply transformed everything! Combined with implementing Number Talks, I believe we have the chance to change the trajectory of math education for our students.

Essentially, the running record is a math interview where we get a measure on the students’ accuracy, flexibility using strategies, and speed (although this part is not nearly as important to me as accuracy and flexibility since I believe, like Jo Boaler’s, that you don’t need to be fast at math to be good at math). Once we figure out where the student falls on the progression, I have been developing a collection of games and activities for each strategy so we can begin working with the student on the strategy they need and help them move forward on the progression. It has been awesome to watch the progress our students are making!

To teach the strategies, we use math tools such as 10 frames, Cuisenaire rods, rekenreks, counting objects and dry erase board 10 frames. Students need to see concretely how and why this strategies work. I have created a set of videos (click here for the webpage) that shows exactly how we are using these tools to teach these strategies. Here is the progression of the strategies we are using:

Plus 1 – We want student to understand that adding one to a number results in the next counting number.

Plus 0 – We tend to combine the Plus 0 examples with the Plus 1 examples since adding 0 is such a difficult concept for our young kiddos.

Count on 2 or 3 – For this strategy, we focus on starting with the larger number especially when the larger number comes second…so 2 + 6 I would want my student to start at the 6 and count on 2. The last thing we want to do is encourage children to count, but if we are just adding 2 or 3 and we have this strategy in place, it will take about the 3 seconds we are looking for to get an accurate answer.

Adding within 5 – We want to be sure that students have a good handle on adding numbers whose sums are up to 5 before we move on the sums within 10 so this is a great checking point.

Adding within 10 – We want to be sure students are comfortable working with sums within 10 before moving on to the next strategy.

Adding to make 10 – The most crucial skill! You can’t practice it often enough even when children pass this strategy. This is a strategy in which we want automaticity. Our students play lots of games like “Go Ten” which is played just like Go Fish but you are asking your partner for the pair that makes 10 with yours or we place 12 cards face up on a table and students are choosing two that combine to make 10. We assess this strategy by giving the students a number and they tell us the number that adds with it to make 10. Before we move a student on from this strategy, we want to be sure that if we say “2” they say “8” and so on without hesitating. This forms such an important foundation for future math concepts.

(disclaimer here – at this point with our K-2 kiddos, we then start them on the subtraction strategies within 10…once they finish those, they reenter this progression here)

Plus 10 – we want our student to develop automaticity with 10 plus any single-digit number since that, too, is a foundational skill embedded in more difficult concepts later on

Doubles – the last of the strategies that we look for automaticity since it is also a foundational skill for future strategies

Doubles plus 1 – We want students to recognize that if they are adding two numbers that are 1 apart, they can double one of the numbers and adjust accordingly. For example, 6 + 7 can be thought of as 6 + 6 + 1 or some children prefer to think of it as 7 + 7 – 1.

Doubles plus 2 – We want students to recognize that if they are adding two numbers that are two apart, they can do Doubles Plus 2, Doubles Minus 2 or even Double the middle. Let’s say we have 6 + 8….Student 1 can think 6 + 6 + 2 = 14, Student 2 could think 8 + 8 – 2 = 14, and Student 3 could think of taking one away from the 8 and giving it to the 6 to change the expression to 7 + 7 = 14. There is no one right strategy..the idea is that children are thinking flexibility about numbers and are using a strategy easiest for them.

Decompose to make 10 when adding a 9 – These last two strategies are the most important strategies because they translate perfectly to larger numbers, adding fractions and decimals, and within concepts of adding units of measurement. Since one of the addends is a 9, we can take away 1 from the other addend and rename the expression with a 10 and whatever is left over. For example, 9 + 6 would be changed into 10 + 5 to make it easier for our brains. Focusing on just the 9 first helps the students develop the ability to think about taking an amount from the other number to make a 10.

Decompose to make 10 when adding a 7 or 8 – This is the last strategy on the progression and forms the foundation for our work with larger numbers down the line. We need to take either 3 from the other number to make a 10 with the 7 or 2 from the other number to make a 10 with the 8.

The decomposing strategy with 7, 8 or 9 is truly the most versatile, so if a child is using this strategy when they are working with any of the ones that fit into doubles plus 1 or doubles plus 2 I would not make students work on the doubles strategies. But, I have found students to stick with the doubles strategy for many expressions such as 5 + 8. Students will say they know 5 + 5 + 3 makes 13. For them, I do want to work with them on decomposing to make a 10 since it is so versatile to future math concepts.

One of the things I feel passionate about is teaching math facts strategically even with our youngest students. Not only are we teaching them strategies for figuring out the particular math facts that are appropriate for their development, but there are two additional bonuses: we are teaching about the conceptual meaning of the operation itself AND we are providing our students with the confidence to try to solve problems they never thought they could (as opposed to students who have learned by memorizing math facts and won’t even attempt more difficult ones because they say they “haven’t learned that one yet”).

Recently I worked with some students to understand the strategy of double, double, double with their multiplication math facts of 8. One of my favorite and, I believe, most powerful math tools is Cuisenaire rods because they help children see that numbers exist as a group rather than a collection ones that need to be counted. I particularly love them to show multiplication problems. We first began by building our rectangle for the problem 8 x 6. My students build their rectangles with the vertical alignment of the first factor and then the horizontal alignment of the second factor (this helps tremendously down the road when we do division). Here’s a picture of it:

There were 8 of the 6 rods, so we built an 8 by 6 rectangle and we then needed to figure out the area inside the rectangle. I wanted to help students find an efficient way for figuring this out rather than just skip counting by six 8 times. So, I had them separate the rods like this:

That way, they could see that the 8 x 6 rectangle has the same area as two 4 x 6 rectangles. We discussed that we can think of any number times 8 as being the same product as a doubling of 4 times that number.

There were a couple of students in my small group who didn’t know that 4 x 6 was 24 so I had them separate their rods like this:

They could see that x4 is the same as x2 doubled. So, the 2×6 has an area of 12, we double that for 4×6 to equal 24 and then we double it again for 8 x 6 for a total of 48. After doing a few of these, the students were getting pretty good at figuring out some products with their x8 math facts. I then asked them if they could figure out what 8 x 16 would be. Within a couple of minutes they had the product of 128. (Down the road I know I’ll be discussing breaking apart the 16 by the place value positions, but this flexibility of thought is exactly what I’m trying to develop.) It was awesome! The students were so proud of themselves. There is no way anyone can memorize all the possible products, but by giving them a strategy for multiplying anything times 8, they were confident they could attempt it. Loved it!

A year and a half ago, I was offered the amazing opportunity to leave my 5th grade classroom of 13 years and become a Title 1 funded Math Coach K-5. I loved having my own classroom and teaching all subjects, but Math has always been my passion and the opportunity to share my passion with all teachers, assistants, parents, and students was simply a dream come true. I had a steep learning curve, though, since I knew nothing about how our youngest children learn math. So I read everything and anything on early numeracy and discovered the most amazing educators in cyberspace!! I was especially enamored and impressed with the educators in MTBoS. I have looked forward to checking my Twitter every day to see what new things I will learn, activities I can try in my classroom, and topics that really get me to think and reflect on my teaching practices. After seeing a particularly persuasive video of an MTBoS member encouraging more people to get involved, I decided that I wanted to become a more active member of this amazing group of educators and signed up for a mentor to help me get going. I couldn’t believe it when I learned that I was assigned Tracy Zager as my mentor since she was one of these amazing educators I was following on Twitter! Crazy cool!

My title “Setting the Foundation” serves a dual purpose with this blog post since it is my first post and, thus, is literally the foundation upon which I will build my blog, but it is also metaphorically a phrase that describes what has become my life’s work! The more that I work with students K-5, the more convinced I am that we need to teach them strategies that will help them succeed not only today but tomorrow as well. When we teach our first graders how to add two single-digit numbers by using strategies like make a ten, doubles plus 1, or bridge 10 to add over ten, we are not only helping them solve the problems they see today, but we are providing the foundation upon which they will progress to learn addition with larger numbers and even fractions and decimals down the line. When we teach 3rd graders how to build concrete models like an open array to show a multiplication problem, we are not only teaching them how to solve a problem today, but we are setting the foundation for using a model that will work with single and multi-digit whole numbers, then decimals in 5th grade, and even polynomials in algebra further on their math journey. It just make so much sense!

One last thing I want to mention in this first post…underlying all my work with teaching math facts strategically, encouraging the development of mental math strategies through Number Talks, and teaching models that will follow students as they progress on their journey, there is one foundational idea that I believe in my heart can change the course of anyone’s math journey – having a growth mindset. Jo Boaler’s ideas are revolutionary and so, so important. They encompass the idea that everyone can learn math! There is no “math brain”…our beliefs about math are shaped by our experiences and attitudes towards math. Our brains have the amazing ability to grow and change and by believing that we CAN learn by trying out new strategies and asking for help when we need it, we CAN learn things that we never dreamed we could learn! Helping children take chances and not be afraid of making mistakes…indeed, as Jo Boaler shares, brain research shows that when we are being challenged and we make a mistake trying out a new strategy, our brain grows! When we realize we made a mistake, it grows again, and when we fix it, it grows again. I want to be a part of this revolution that changes people’s mindsets about math and help those who have developed a phobia about math to see that math is beautiful and accessible to all! Thanks so much for joining me on this journey!